Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.9 Representations of Functions as Power Series - 11.9 Exercises - Page 797: 18


$\sum_{n=0}^{\infty}n(n-1)\frac{x^{n+1}}{2^{n+2}}$, $R=2$

Work Step by Step

$f(x)=(\frac{x}{2-x})^{3}=\sum_{n=0}^{\infty}n(n-1)\frac{x^{n+1}}{2^{n+2}}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{(n+1)((n+1)-1)\frac{x^{n+2}}{2^{n+3}}}{n(n-1)\frac{x^{n+1}}{2^{n+2}}}|$ $=|\frac{x}{2}|\lt 1$ The given series converges with $R=2$
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